Seventh derivative at zero of a polynomial times a Gaussian

Define $f(x)=e^{x^2}\left(3x^3+2x^2+x+1\right)$. What is the seventh derivative $f^{(7)}(0)$?

$8400$. $f^{(7)}(0)=7!\,[x^7]f$. Since $e^{x^2}=1+x^2+\tfrac{x^4}{2}+\tfrac{x^6}{6}+\cdots$ has only even powers, $x^7$ comes from $x\cdot\tfrac{x^6}{6}$ and $3x^3\cdot\tfrac{x^4}{2}$, giving coefficient $\tfrac16+\tfrac32=\tfrac53$ and $5040\cdot\tfrac53=8400$.